Automorphism Groups of Kausz Compactifications

• Generalized Kausz compactifications are moduli spaces constructed via canonical blow-ups on Grassmannians, providing explicit control over automorphism and Picard groups.
• The construction employs blow-ups along loci determined by vanishing Plücker coordinates, yielding boundary divisors in simple normal crossings configuration.
• Automorphism group classification uncovers key symmetries, including USD and DUAL involutions, linking classical group actions with modern moduli theory.

Generalized Kausz compactifications, denoted $\mathcal T_{s,p,n}$, are moduli-theoretic compactifications constructed from the Grassmannian and equipped with natural boundary divisors arising via canonical blow-up procedures. The automorphism groups of these varieties, and of related spaces of complete collineations, reveal deep connections between linear algebraic group actions, the geometry of Grassmannians, and intersection theory on spherical varieties. The theory extends classical constructions, like the wonderful compactification of $GL_p$, and offers explicit control over automorphism groups, Picard groups, and anticanonical divisor positivity properties, providing a uniform geometric framework across a broad class of parameters $(s,p,n)$ (Fang, 6 Jan 2026). All work assumes an algebraically closed field $\mathbb K$ of characteristic zero.

1. Construction of $\mathcal T_{s,p,n}$

Let $E$ be an $n$-dimensional vector space over $\mathbb K$, with a direct sum splitting $E = E_1 \oplus E_2$, $\dim E_1 = s$, $\dim E_2 = n-s$. The Grassmannian $G(p, E)$ of $p$-planes in $E$ admits a Plücker-type embedding into $\mathbb P(\wedge^p E)$. The exterior power decomposes as

$$\wedge^p E \cong \bigoplus_{k=0}^p \wedge^k E_1 \otimes \wedge^{p-k} E_2,$$

yielding a rational map

$$\mathcal K_{s,p,n}\colon G(p, E) \dashrightarrow \mathbb P(\wedge^p E) \times \prod_{k=0}^p\mathbb P(\wedge^k E_1 \otimes \wedge^{p-k} E_2).$$

The generalized Kausz compactification is defined as the Zariski closure of the graph of this map: $$\mathcal T_{s,p,n} := \overline{\operatorname{Graph}(\mathcal K_{s,p,n})} \subset G(p, E) \times \prod_{k=0}^p\mathbb P(\wedge^k E_1 \otimes \wedge^{p-k} E_2).$$ Alternatively, $\mathcal T_{s,p,n} \to G(p, E)$ is obtained by a canonical sequence of blow-ups at explicitly determined loci

$$S_k := \big\{x \in G(p, E)\ \mid\ \text{all Plücker coordinates in}~\wedge^k E_1 \otimes \wedge^{p-k} E_2~\text{vanish}\big\},\ 0 \leq k \leq r:=\min\{s, n-s, p, n-p\}.$$

Blowing up $G(p, E)$ along $S_0$, then the strict transform of $S_1$, and so forth up to $S_r$ yields a variety isomorphic to $\mathcal T_{s,p,n}$; the construction is independent of the order of the $S_k$.

The boundary consists of $2r$ smooth prime divisors, denoted $D_1^-, \ldots, D_r^-, D_1^+, \ldots, D_r^+$, in simple normal crossings configuration. The Picard group is freely generated by the pullback $H$ of the hyperplane class on $G(p, E)$ and these boundary divisors, except in certain low-rank cases. The effective cone is spanned by the $2r$ boundary divisors and a small number of strict transforms $B_j$ of the $S_j$ ("B-stable" divisors).

2. Classification of Automorphism Groups of $\mathcal T_{s,p,n}$

The natural group $G = GL(s, \mathbb K) \times GL(n-s, \mathbb K)$ acts equivariantly on $\mathcal T_{s,p,n}$ via its canonical linear action on $\wedge^p E$ and the submodules $\wedge^k E_1 \otimes \wedge^{p-k} E_2$. The automorphism group classification, established in Theorem 1.7, is as follows:

Parameter regime	Automorphism group	Notable symmetry/involution
$n \neq 2s,\, n \neq 2p$	$(GL(s)\times GL(n-s))/Z(GL(n))$	None
$n=2s\neq 2p$	$([GL(s)\times GL(s)]/Z)\rtimes (\mathbb Z/2)$	USD: exchange $E_1/E_2$
$n=2p\neq 2s$	$([GL(s)\times GL(n-s)]/Z)\rtimes (\mathbb Z/2)$	DUAL: Grassmann duality
$n=2s=2p$	$$([GL(s)\times GL(s)]/Z)\rtimes (\mathbb Z/2\rtimes \mathbb Z/2)$$	Both USD and DUAL
Low-rank degenerate cases	Classical projective/parabolic automorphism groups	Degenerate phenomena

Here, $Z$ denotes the center appropriate to the context (typically the scalar matrices in $GL(n, \mathbb K)$), and the USD and DUAL involutions arise respectively from summand exchange and Grassmann duality isomorphisms.

3. Proof Methodology and (Semi-)Positivity of the Anticanonical Bundle

The proof utilizes three main ingredients:

1. Action on the Picard group: Since automorphisms must permute the finite set of boundary divisors and B-stable divisors, and the Picard group is generated by these along with $H$, only involutions corresponding to USD and DUAL manifest as genuine automorphisms affecting Pic$\ \mathcal T$ nontrivially.
2. Descent to $G(p,E)$: Any automorphism fixing $H$ and acting compatibly with the boundary stratification must descend to an automorphism of $G(p,E)$. The automorphism groups of $G(p,E)$ are classical: $PGL(n)$ for $n\neq 2p$ and $PGL(n)\rtimes \mathbb Z/2$ for $n=2p$ (incorporating Grassmann duality).
3. Intersection-theoretic positivity: Intersection numbers with $T$-invariant curves (parametrized via Mille–Crêpes charts) show that $-K_{\mathcal T}$ is nef and big, with ampleness only for $r\leq 2$. Brion's theory of spherical varieties ensures that the cone of effective cycles is generated by the $B$-orbit closures, which suffices to establish the minimality of the automorphism group beyond the expected involutions.

4. Automorphism Groups of Spaces of Complete Collineations $\mathcal M_{s,p,n}$

The related moduli space $\mathcal M_{s,p,n}$, referred to as the space of complete collineations, is constructed as the projection of $\mathcal T_{s,p,n}$: $$\mathcal M_{s,p,n} := \operatorname{Im}\left(\mathcal T_{s,p,n} \to \prod_{k=0}^p\mathbb P(\wedge^k E_1 \otimes \wedge^{p-k} E_2)\right).$$ Equivalently, it is the blow-down of $D_1^+$ in $\mathcal T$, or can be realized as an iterated blow-up of $\mathbb P(N_{G(p,s)/G(p,n)})$, yielding a compactification of the space of rank-$p$ linear maps $E_1 \to E_2$.

The same group $GL(s)\times GL(n-s)$ acts on $\mathcal M_{s,p,n}$, and intersection-theoretic calculations show that $-K_{\mathcal M}$ is ample (unlike $-K_{\mathcal T}$, which is only nef for $r>2$). The automorphism group admits a parallel description, with $GL(\cdot)$ replaced by $PGL(\cdot)$:

• If $n\neq 2s,2p$, $$\operatorname{Aut}(\mathcal M) = PGL(s)\times PGL(n-s)$$.
• If $n=2s\neq 2p$, a USD involution is present.
• If $n=2p\neq 2s$, a DUAL involution is present.
• If $n=2s=2p$, both involutions occur.

5. Corollaries and Explicit Examples

Several special cases of the construction recover notable classical geometries:

• For $(s,p,n) = (p,p,2p)$, $\mathcal T$ and $\mathcal M$ recover the classical Kausz compactification of $GL_p$, and the boundary exceptional divisors encode the wonderful compactification structure; both USD and DUAL involutions are present, giving up to $(\mathbb Z/2)^2$ extra automorphism factors.
• For $p=1$ or $p=n-1$, $\mathcal T_{s,1,s+1} \cong \mathbb P^s$ or $\mathbb P^{n-s}$, and the automorphism group is $PGL$ as expected.
• For $p=1$, $n=2s$, the involution exchanges the two Grassmann factors $\mathbb P^{s-1}\times\mathbb P^{s-1}$.
• For $r=1$ or $2$ (rank of the construction), the anticanonical bundle is ample, therefore both $\mathcal T$ and $\mathcal M$ are Fano.
• The Picard group in all cases has the basis $H, D_1^\pm, \ldots, D_r^\pm$ (with only the appropriate combinations surviving to $\mathcal M$ after blow-down).

6. Significance and Structural Uniformity

These results establish the extent to which wonderful-type compactifications and their automorphism groups can be extended from the linear group setting to moduli spaces with more general block structures. The explicit blow-up constructions from $G(p, n)$, in conjunction with combinatorial (via Mille–Crêpes charts) and intersection-theoretic methods, yield a comprehensive and highly uniform description of automorphism groups and Picard groups for the entire family of generalized Kausz and complete collineation compactifications, with degeneracies arising only in tractable, low-dimensional scenarios (Fang, 6 Jan 2026).